## Abstract

Let G = (V, E) be a graph and let S ⊆ V. The set S is a packing in G if the vertices of S are pairwise at distance at least three apart in G. The set 3 is a dominating set (DS) if every vertex in V - S is adjacent to a vertex in S. Further, if every vertex in V - S is also adjacent to a vertex in V - S. then S is a restrained dominating set (RDS). The domination number of G. denoted by γ(G). is the minimum cardinality of a DS of G, while the restrained domination number of G, denoted by γ_{r}(G), is the minimum cardinality of a RDS of G. The graph G is γ-excellent if every vertex of G belongs to some minimum DS of G. A constructive characterization of trees with equal domination and restrained domination numbers is presented. As a consequence of this characterization we show that the following statements are equivalent: (i) T is a tree with γ(T) = γ_{r}(T); (ii) T is a γ-excellent tree and T ≠ K_{2}; and (iii) T is a tree that has a unique maximum packing and this set is a dominating set of T. We show that if T is a tree of order n with ℓ leaves, then γ_{r}(T) ≤ (n + ℓ+ 1)/2, and we characterize those trees achieving equality.

Original language | English |
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Pages (from-to) | 597-607 |

Number of pages | 11 |

Journal | Journal of Global Optimization |

Volume | 34 |

Issue number | 4 |

DOIs | |

Publication status | Published - Apr 2006 |

Externally published | Yes |

## Keywords

- Domination
- Domination excellent trees
- Restrained domination

## ASJC Scopus subject areas

- Computer Science Applications
- Management Science and Operations Research
- Control and Optimization
- Applied Mathematics